Ed Miliband has sought to brush off questions about his suitability to lead the country, insisting he is "more personally competent" than David Cameron to be prime minister.

The Labour leader played down the significance of a set of polls showing that the Conservatives had established a lead over his party for the first time since George Osborne's "omnishambles" budget more than two years ago.

Two surveys – an ICM poll for the Guardian and another carried out for the Tory peer Lord Ashcroft – gave Cameron's party a two-point lead a year ahead of the 2015 general election.

The Guardian poll put Labour support at 31% – its lowest since the immediate aftermath of the 2010 election – with Miliband's personal ratings falling well behind those of the prime minister and the chancellor.

Just over a week before elections to English councils and the European parliament on 22 May, Miliband insisted he was right to focus on the "cost-of-living crisis" facing British families.

Despite positive economic figures over recent months, ordinary people were "deeply discontented" with how the UK was being run, and felt that the benefits from growth had been skewed towards those at the top of society, he said.

Challenged over whether voters saw him as a potential prime minister, Miliband told BBC1's Breakfast: "My approach has been to talk consistently about the big questions our country faces and the biggest question of all that every country is wrestling with: are we going to be run for a few people at the top, with growing gaps between the richest and everybody else, or are ordinary people going to get a fair shot?

"I am going to keep talking consistently about that, because I think that is the biggest question our country faces."

Asked if this approach would make him a better prime minister than Cameron, he said: "It makes me more personally competent because I get what people are feeling and we can do something about it.

"The most important thing as a prime minister is to understand what people think and what you can do about it and to show the difference you can make to their lives, and that's what I will do, and solidity of belief and purpose."

The shadow health secretary, Andy Burnham, said next week's elections would be a better test than opinion polls of whether Labour's policies "make sense to people".

"Polls will narrow, won't they, as we get towards a general election. That is what always happens," he told BBC Radio 4's Today programme.

"Labour has a lead in one of the polls this morning. The important thing actually though is not what the polls are saying. The important thing is: have Labour got the answers to the problems that people are facing?

"Ed Miliband has spoken about measures to take the pressure off families: help with energy bills; controls on the private rented sector. These are the things that matter to people. We'll see next Thursday, won't we, whether those policies are making sense to people and giving them real answers to the problems they face."

He said critics of the broadcast, filmed like a 1950s B-movie, should "lighten up a bit. We were talking there about the substance of what this government has done," he told Breakfast. "Let's have a debate about whether it is fair to have a bedroom tax on ordinary families while cutting taxes on the richest in society. I don't think that's fair. I don't think it's right. I don't think it's what a One Nation government should do."

Miliband challenged the idea that improvements in GDP were winning voters over to the Tories: "I don't really see it that way. The government says the economy is fixed and everything is fine. What I see going out and talking to people – and some of the polls are reflective of this – is deep discontent with the way the country is being run.

"This is not a country where people think everything is going fine. I think this is a country where people are deeply frustrated about the way the country is run. They really feel they are not getting a fair shot."

Miliband was given a net approval rating of minus 25, with 51% of voters believing he is doing a bad job and 26% believing he is doing well. Cameron enjoyed a dramatic improvement from the minus 15 rating he received 12 months ago. He is now on plus two, with 44% of voters believing he is doing a good job, against 42% who think he is doing badly.

The first in a series of weekly telephone surveys carried out for Lord Ashcroft put Cameron's party on 34%, two points ahead of Labour on 32%, with Ukip third on 15% and the Liberal Democrats trailing on 9%.

Miliband said: "Polls go up and down. I've seen that over three and a half years in this job. I think what matters is talking about the bread and butter issues people face – energy prices, childcare, the NHS, how we can improve GP access, something Labour is leading the other parties on.

"We will talk about the issues that matter to people. We will focus on these questions. The people will make the decisions a week on Thursday and at the general election."